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Iterations

Calculate
algebraic series
such as e = 1+ 1/2! + 1/3! + ..., a square wave,Fibonacci numbers.
Study iterative maps, e.g. the (one-dimensional)
Logistic map:
(see below) or more complicated multi-dimensional maps. The logistic map is perhaps one of the simples mathematical system showing many characteristics
of the development of
chaotic
behaviour. Several analytical tools
are valailable to study the results of Iterations and ODE's such as:

A special window has been added in Mathgrapher v2 to allow detailed presentation
of 2D orbits at the pixel level and to study the stability of the orbits (see the Examples and Demonstrations for the Henon map, Standard map and Mandelbrot and Julia sets.

Choose a pair of F's from the Functions that are iterated and push Draw to make a
graph where the X-axis represents the first and the Y-axis the second Function.

The example below is a graph of F1 and F2 for a series of 1024 iterations of the logistic map.
The first 100 iterations are omitted. Note that a 4-cycle has developed for a=0.87.

Same as above for a=0.89 showing an 8-cycle and a=0.96 (the gap in the bifurcation diagram)
where we have a 3-cycle. The behaviour is chaotic above 0.96.